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10
P
with
λ∈R
,where
g:[
0,1
]→R
,
k:[
0,1
]×[
0,1
]→R
and
f:R→R
aregivenfunctions,andthefunction
x:[
0,1
]→R
isunknown.Using
theoperators
(3)
and
(4)
,wemayrewrite
(6)
astheoperatorequation
x=g+λK(Cf(x)),
(7)
andthestructureof
(7)
suggeststoapplyfixedpointprinciples.More
generally,theHammersteinequation
x(t)=g(t)+λ∫
0
1
k(t,s)f(s,x(s))ds
(0<t<1),
(8)
wherenow
f:[
0,1
]×R→R
dependsalsoon
s
,leadstothefixedpoint
equation
x=g+λK(Sf(x))
(9)
involvingthesuperpositionoperator
(5)
.Occasionally,wewillalso
considerHammerstein–Volterraequations,wheretheupperintegration
limit1in(6)and(8)isreplacedbythevariablelimitt.
Itiswell-knownthatHammersteinequationsnaturallyoccurin
thestudyofboundaryvalueproblems,whileHammerstein–Volterra
equationsnaturallyoccurinthestudyofinitialvalueproblems.So
everyexistence(anduniqueness)resultweobtainfortheoperator
equations
(7)
and
(9)
leadstoacorrespondingexistence(andunique-
ness)resultforboundaryorinitialvalueproblems.Examplesofsuch
problemsarebrieflydiscussedinChapter5.Forthereaders’easewe
haveaddedalistoftablesaswellassymbolandsubjectindicesat
theend.
Wedohopethatreaderswhoarenotexpertsinthetheoryand
applicationsoffunctionsofboundedvariationbutwanttogetanidea
ofthedevelopmentsinthelastdecades,aswellasaglimpseofthe
diversityinwhichcurrentresearchismoving,willfindthissurvey
bothreadableandstimulating.
Acknowledgements0
Thissurveyistheoutcomeofseveralmeetings
andfruitfuldiscussionsoftheauthors.Ourprofoundgratitudegoesto
theMathematicalResearchInstituteinOberwolfach,Germany,where
theauthorsspentacoupleofweeksintheframeworkofthe“Research
